dynamics of uniform circular motionsrjcstaff.santarosa.edu/~lwillia2/20/20ch5_s17.pdf ·...
TRANSCRIPT
-
© 2010 Pearson Education, Inc.
Chapter 5
Dynamics of Uniform Circular
Motion
Circular Motion, Orbits,
and Gravity
-
© 2010 Pearson Education, Inc. Slide 6-3
-
© 2010 Pearson Education, Inc. Slide 6-5
-
Circular Motion
-
Milky Way Galaxy
Orbital Speed of Solar System: 220 km/s
Orbital Period: 225 Million Years
-
Mercury: 48 km/s
Venus: 35 km/s
Earth: 30 km/s
Mars: 24 km/s
Jupiter: 13 km/s
Neptune: 5 km/s
-
Equatorial Rotational Period: 27.5 days
Differential Solar Rotation
Entangled Magentic Field
Lines makes Sun Spots
-
464m/s
-
Precession
causes the
position of the
North Pole to
change over a
period of
26,000 years.
-
Orbital Speed of Earth: ~ 30 km/s
-
Although the
Moon is always
lit from the
Sun, we see
different
amounts of the
lit portion from
Earth
depending on
where the
Moon is
located in its
month-long
orbit.Orbital Speed of Moon: ~ 1 km/s
-
155mph~70m/s
-
RADAR: RAdio Detecting And Ranging
200mph~90m/s
-
Electrons in Bohr Orbit: 2 million m/s
(Speed of Light: 200 million m/s)
-
Maximal Kerr Black Hole
Rotational Speed = Speed of Light
-
Uniform Circular Motion
The velocity vector is tangent to the path
The change in velocity vector is due to the change in direction.
The centripetal acceleration changes the direction of motion:
va
t
-
Period & Frequenccy of
Rotation
What is the period of the Earth? Moon?
-
Centripetal Acceleration
• The acceleration is always
perpendicular to the path of the motion
• The acceleration always points toward
the center of the circle of motion
• This acceleration is called the
centripetal acceleration
• Magnitude is given by:2
C
va
r
-
© 2010 Pearson Education, Inc.
Circular Motion: centripetal acceleration
There is an acceleration
because the velocity is
changing direction.
Slide 3-43
-
Centripetal and Centrifugal Center SEEKING and Center FLEEING
Factual = Centripetal Force
Ffictitious = Centrifugal Force
-
In Physics, we use
ONLY
CentriPETAL acceleration
NOT
CentriFUGAL acceleration!
-
Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.
5.3 Centripetal Force
Example 5: The Effect of Speed on Centripetal Force
The model airplane has a mass of 0.90 kg and moves at
constant speed on a circle that is parallel to the ground.
The path of the airplane and the guideline lie in the same
horizontal plane because the weight of the plane is balanced
by the lift generated by its wings. Find the tension in the 17 m
guideline for a speed of 19 m/s.
r
vmTFc
2
N 19m 17
sm19kg 90.0
2
T
-
The Earth rotates once per day around its axis as shown.
Assuming the Earth is a sphere, is the rotational speed at Santa
Rosa greater or less than the speed at the equator?
366 m/s
464 m/s
-
The Earth rotates once per day around its axis. Assuming the Earth is a sphere
with radius 6.38 x 106m, find the tangential speed of a person at the equator
and at 38 degrees latitude (Santa Rosa!) and their centripetal accelerations.
At the equator, r = 6.38 x 106m:
62 2 (6.38 10 )464 /
86,400
r x mv m s
t s
22
2
6
464 /.034 /
6.38 10 c
m sva m s
r x m
At Santa Rosa, r = 6.38 x 106m cos38:
62 2 (6.38 10 )cos38366 /
86,400
r x mv m s
t s
22
2
6
366 /.027 /
cos38 6.38 10 cos38 c
m sva m s
r x m
ca
ca
-
Is your apparent weight as
measured on a spring scale more
at the Equator or at Santa Rosa?
ca
g
-
Since you are standing on the Earth
(and not in the can) the centrifugal force tends
to throw you off the Earth. You weigh less
where the centripetal force is greatest because
that is also where the centrifugal force is
greatest – the force that tends to throw you
out of a rotating reference frame.
-
Centripetal & Centrifugal Force
Depends on Your Reference Frame
Inside Observer
(rotating reference frame)
feels Centrifugal Force
pushing them against the
can.
Outside Observer
(non-rotating frame) sees
Centripetal Force pulling
can in a circle.
Center-seeking
Center-fleeing
-
Centrifugal Force is Fictitious?The centrifugal force is a real
effect. Objects in a rotating frame
feel a centrifugal force acting on
them, trying to push them out. This
is due to your inertia – the fact that
your mass does not want to go in a
circle. The centrifugal force is
called ‘fictitious’ because it isn’t
due to any real force – it is only due
to the fact that you are rotating.
The centripetal force is ‘real’
because it is due to something
acting on you like a string or a car.
-
© 2010 Pearson Education, Inc.
Important: Inside vs Outside the Rotating Frame
-
5.6 Apparent Weightlessness and Artificial Gravity
Example 13: Artificial Gravity
At what speed must the surface of the space station move
so that the astronaut experiences a push on his feet equal to
his weight on earth? The radius is 1700 m.
mgr
vmFc
2
s/m130
sm80.9m 1700 2
rgv
-
5.6 Apparent Weightlessness and Artificial Gravity
Example 13: Artificial Gravity
At what speed must the surface of the space station move
so that the astronaut experiences a push on his feet equal to
his weight on earth? The radius is 1700 m.
s/m130v
What is the period of rotation?
-
Space Station Rotation
A space station of d iameter 80 m is turning about
its axis at a constant rate. If the acceleration of the
outer rim of the station is 2.5 m/ s2, what is the
period of revolution of the space station?
a. 22 s
b. 19 s
c. 25 s
d. 28 s
e. 40 s
2 2, C
v ra v
r
-
© 2010 Pearson Education, Inc.
Horizontal (Flat) Curve
The force of static friction supplies the centripetal force
The maximum speed at which the car can negotiate the curve is
Note, this does not depend on the mass of the car
v gr
2
cmv
F fr
0 yF N mgf mg
-
© 2010 Pearson Education, Inc.
A highway curve has a radius of 0.14 km and is
unbanked. A car weighing 12 kN goes around the
curve at a speed of 24 m/s without slipping. What
is the magnitude of the horizontal force of the
road on the car? What is μ? Draw FBD.
a. 12 kN
b. 17 kN
c. 13 kN
d. 5.0 kN
e. 49 kN
Horizontal (Flat) Curve
-
© 2010 Pearson Education, Inc.
Example Problem
A level curve on a
country road has a
radius of 150 m. What
is the maximum speed
at which this curve can
be safely negotiated on
a rainy day when the
coefficient of friction
between the tires on a
car and the road is
0.40?
Slide 6-28
-
Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.
5.4 Banked Curves
On a frictionless banked curve, the centripetal force is the
horizontal component of the normal force. The vertical
component of the normal force balances the car’s weight.
-
Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.
5.4 Banked Curves
r
vmFF Nc
2
sin mgFN cos
-
Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.
5.4 Banked Curves
r
vmFN
2
sin
mgFN cosrg
v2tan
-
Copyright © 2015 John Wiley & Sons, Inc. All rights reserved.
5.4 Banked Curves
Example 8: The Daytona 500
The turns at the Daytona International Speedway have a
maximum radius of 316 m and are steeply banked at 31
degrees. Suppose these turns were frictionless. At what
speed would the cars have to travel around them in order
to remain on the track?
rg
v2tan tanrgv
mph 96 sm4331tansm8.9m 316 2 v
-
© 2010 Pearson Education, Inc.
Vertical Circle has Non-Uniform Speed
Where is the speed Max? Min?
Where is the Tension Max? Min?
-
© 2010 Pearson Education, Inc.
Example Problem: Loop-the-Loop
A roller coaster car goes through a vertical loop at a constant
speed. For positions A to E, rank order the:
• centripetal acceleration
• normal force
• apparent weight
Slide 6-32
-
© 2010 Pearson Education, Inc.
What is the maximum speed the vehicle can have at B and still remain on the track?
-
© 2010 Pearson Education, Inc.
:
n
mg
Humps in the Road
: 0Take n
v gr
What is the maximum speed the car can have as it passes this highest point without losing contact with the road?
Max speed without losing contact MEANS:
2mv
mgr
Therefore:
Maximum Speed for Vertical
Circular Motion
Maximum Speed to not loose contact with road only depends on R!
ROOT GRRRRRRRR
-
© 2010 Pearson Education, Inc.
What is the maximum speed the vehicle can have at B and still remain on the track?
v gr
-
© 2010 Pearson Education, Inc.
A car that’s out of gas coasts
over the top of a hill at a steady
20 m/s. Assume air resistance
is negligible. Which free-body
diagram describes the car at
this instant?
QuickCheck 8.10
Slide 8-80
Humps in the Road: Outside the Vertical Loop
-
© 2010 Pearson Education, Inc.
A car that’s out of gas coasts
over the top of a hill at a steady
20 m/s. Assume air resistance
is negligible. Which free-body
diagram describes the car at
this instant?
QuickCheck 8.10
Now the centripetal
acceleration points down.
Slide 8-81
Humps in the Road: Outside the Vertical Loop
-
© 2010 Pearson Education, Inc.
A roller coaster car does a loop-
the-loop. Which of the free-body
diagrams shows the forces on
the car at the top of the loop?
Rolling friction can be neglected.
QuickCheck 8.11
Slide 8-82
Loop d’ Loops: Inside the Vertical Loop
-
© 2010 Pearson Education, Inc.
A roller coaster car does a loop-
the-loop. Which of the free-body
diagrams shows the forces on
the car at the top of the loop?
Rolling friction can be neglected.
QuickCheck 8.11
The track is above the car, so
the normal force of the track
pushes down.Slide 8-83
Loop d’ Loops: Inside the Vertical Loop
-
© 2010 Pearson Education, Inc.
A roller-coaster car has a mass of 500 kg when fully loaded with passengers. At the bottom of a circular dip of radius 40 m (as shown in the figure) the car has a speed of 16 m/s. What is the magnitude of the force of the track on the car at the bottom of the dip?
a. 3.2 kN
b. 8.1 kN
c. 4.9 kN
d. 1.7 kN
e. 5.3 kN
Loop d’ Loops: Inside the Vertical Loop
-
© 2010 Pearson Education, Inc.
Orbital Mechanics
-
Projectile Motion/Orbital Motion
Projectile Motion is Orbital motion that hits the Earth!
-
Curvature of Earth
If you threw the ball at 8000 m/s off the surface of the Earth
(and there were no buildings or mountains in the way)
how far would it travel in the vertical direction in 1 second?
Curvature of the Earth: Every 8000 m, the Earth curves by 5 meters!
The ball will achieve orbit.
2 2 21 ~ 5 / (1 ) 52
y gt m s s m
-
Orbital VelocityIf you can throw a ball at 8000m/s, the Earth curves away
from it so that the ball continually falls in free fall around the
Earth – it is in orbit around the Earth!
Above the atmosphere
Ignoring
air
resistance.
-
Orbital Motion| & Escape Velocity
8km/s: Circular orbit
Between 8 & 11.2 km/s: Elliptical orbit
11.2 km/s: Escape Earth
42.5 km/s: Escape Solar System!
-
5.5 Satellites in Circular Orbits
There is only one speed that a satellite can have if the
satellite is to remain in an orbit with a fixed radius.
2
s Em M GFr
2va
r
sF m a
-
Orbit QuestionFind the orbital speed of a satellite 200 km above the Earth.
Assume a circular orbit.
2va
r
2
s Em M GFr
sF m a
2
2
s Es
m M G vF m
r r E
E
M Gv
R h
24 11 2 2
6
(5.97 10 )(6.67 10 / )
6.58 10
x kg x Nm kgv
x m
37.78 10 /v x m s
24 65.97 10 , 6.38 10E EM x kg R x m
Notice that this
is less 8km/s!
What is this?
-
Orbit Question
2 r
v
What is the period of a satellite orbiting 200 km above the Earth?
Assume a circular orbit.
2 rv
5314 88mins
24 65.97 10 , 6.38 10E EM x kg R x m
6
3
2 (6.58 10 )
7.78 10 /
x m
x m s
2
2
GMm vF m
rr
2(2 / )rm
r
22 34 r
GM
Kepler’s 3rd!
If you don’t know the velocity:
Period increases with r!
-
g and v Above the Earth’s Surface
If an object is some
distance h above the
Earth’s surface, r
becomes RE + h 2
E
E
GMg
R h
The tangential speed of an
object is its orbital speed
and is given by the
centripetal acceleration, g:
2vg
r
2
E
v
R h
( )
E
E
GMv
R h
2
E
E
GM
R h
2
GMmF
r
Orbital speed decreases with
increasing altitude!
-
5.5 Satellites in Circular Orbits
Global Positioning System
hours 24T
EGM
rT
232
What altitude for a
Geosynchronous orbit?
-
© 2010 Pearson Education, Inc.
Additional Questions
A satellite orbits the earth. A Space Shuttle crew is sent to boost the
satellite into a higher orbit. Which of these quantities increases?
A. Speed
B. Angular speed
C. Period
D. Centripetal acceleration
E. Gravitational force of the earth
Slide 6-43
-
© 2010 Pearson Education, Inc.
Answer
A satellite orbits the earth. A Space Shuttle crew is sent to boost the
satellite into a higher orbit. Which of these quantities increases?
A. Speed
B. Angular speed
C. Period
D. Centripetal acceleration
E. Gravitational force of the earth
Slide 6-44
-
Types of Orbits
-
1071 operational satellites in orbit around the Earth. 50 percent of which
were launched by the United States.Half of that 1071 are in Low-Earth
Orbit, just a few hundred kilometers above the surface.
-
https://www.youtube.com/watch?v=3AzhmNQfVR8
https://www.youtube.com/watch?v=3AzhmNQfVR8https://www.youtube.com/watch?v=3AzhmNQfVR8
-
NASA EARTH
OBSERVATION
-
https://www.youtube.com/watch?v=eYVsVRgiS0w
https://www.youtube.com/watch?v=eYVsVRgiS0whttps://www.youtube.com/watch?v=eYVsVRgiS0w
-
War in Space